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Volatility Project - Group 4
Exploring Types of Volatility and Its Use in Investments
December 14, 2007
Derek Light Chiao Liu
Elizabeth Martin Yuliya Ostrovska
Roger Obourn Tiffany Perkins
Brian Peterson Todd Peterson
This report was completed as a group assignment for Finance 622 with Dr. Arlyn Rubash as part
of the MBA Program at Bradley University.
LEGAL DISCLAIMER*WARNING* - This information should not be construed as adviceor recommendations for any investing or trading activity. Investment and trading activities are
risky by nature, and investors should seek knowledgeable and reliable counsel before engaging
in any type of trading activity.
FoldTable of Contents
IntroductionBlack-Scholes Option Pricing ModelTypes of VolatilityHistorical Volatility - HVImplied Volatility - IVVolatility IndexesVIX - CBOE Volatility IndexVXN - CBOE NASDAQ-100 Volatility Index
VXD - CBOE DJIA Volatility IndexVIX, VXN , VXD Option StrategiesConclusionBibliography
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Introduction
Today, many people understand the basics of investments and have the ability to follow financial
news to stay informed on market moves and stock price changes. Since financial information is
readily available all the time through websites such as Yahoo! Finance and the Wall StreetJournal Online, accessing this news is now easier than ever. Through these sites, anyone can find
the price histories of securities such as bonds, stocks, Treasury securities and various derivatives.
This report provides a look into the practice of using Historical Volatility data and Implied
Volatility data to measure volatility in order to predict performance. We will analyze these
different methodologies for measuring volatility and determine which method more accurately
forecasts actual volatility under certain conditions.
Black-Scholes Option Pricing Model
Accurately valuing options is difficult. The varying price of the underlying stock, as well as the
timing of when the option is exercised, all affect the value. While it is known that these factors
affect the value, the difficult part is determining by how much. "'We know it is worth less than a
share of stock, and more than zero,' says Tim Lucas, former research director at the Financial
Accounting Standards Board,"1
One of the standard finance models for predicting a stock price change can be described with the
Black-Scholes model. In 1973 Fisher Black and Myron Scholes developed what is now coined
the Black-Scholes model, which is a very common derivative-based model used to value optionsand is the foundation of option pricing. An option is a contract, which gives the buyer the right,
but not an obligation, to buy or sell an underlying asset at a specific price on or before a certain
date. An option to buy is known as a call and an option to sell is a put. "Indeed, some bankers
argue that the Black-Scholes theory, which provides an easy way to value options, has had as
much impact on finance as the discovery of DNA has had on medicine."2The Black-Scholes
method is based on the early works of Louis Bachelier and the Geometric Brownian Motion
Model. The graph below illustrates the model of stock price as a function of time:
(1)
$dS/S$=$$dt+$$dz
where
dS/S= the instantaneous rate of return on a stock dt$ = the change in the stock's return over time dt and dz$ = describes the uncertainty, volatility and drift3
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Figure 1: Stock Price as a Function of Time
4Geometric Brownian Motion Model
The model requires five components to effectively value an option:
Price of the underlying stock Strike price Time in years until expiration Risk-free interest rate Standard deviation5
Of the five components of the formula, T or time until maturity is expressed in terms of years
and must be scaled at an annualized basis. For example, a maturity of 9 months is the equivalent
of 9/12 or .75 and is the value used in the formula to accurately account for time when valuing an
option. To understand whether an option is an ideal agreement to enter, both the present value of
the strike price and current price of the underlying asset are required. This supports the notion
that you must know what you have and where you have been in order to understand where you
are going. The same logic applies to the current price and strike price. The strike price, or
exercise price, is the agreed upon price for a call or put at the expiration of the option. The
standard deviation, or volatility component, is a measurement of variability over time. The term
risk-free interest rate is described as the offered interest rate minus non-environmental risk.
Environmental risk must be factored in because businesses operate in an uncertain environment
and are part of an open system.
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Black-Scholes formula:
(2)
c0=S0N(d1)PV(K)N(d1T)
where
(3)
d1=ln(S0PV(K))T+T2
and
C0 = call premium
S0 = current stock price
N() = normal distributionPV= present value
K= strike or exercise price
= standard deviation
The formula provides a lot of information about a stock and helps identify the outcome of any
fluctuations in the five components of the equation. Without solving for the full equation the left
and right-hand portions of the formula provide critical information to an investor, while the full
equation calculates the cost of the tracking portfolio. The left-hand side of the equation yields the
number of shares of stock and the right side represents the number of dollars borrowed at the
risk-free rate.6
The Black-Scholes model is a continuous-time valuation model that allows you to calculate an
infinite number of option prices at any point in time.7In addition to following the Geometric
Brownian Motion Model as noted earlier, the Black-Scholes model assumes that the market is
frictionless, which among other things means:
Continuous trading is possible No transaction costs The purchase of fractional shares is possible Continuous borrowing and lending at the risk-free interest rate is possible Short selling is possible No chance for arbitrage The stock does not pay a dividend8
While 80% of companies use the Black-Scholes method, it is surprising that the number of large
companies using the binomial method for valuing options is as large as 20%. A 2006 study
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by Compliance Weekillustrated that 80% of companies offering employee stock options and
reporting sales over $20 billion use the Black-Scholes method. The other 20% use the binomial
method, which makes sense since the big players are the only ones that can afford the costs
associated with this complicated method. Of the 20% using the binomial method, half of them
actually switched from the Black-Scholes. These companies were Caterpillar, Citigroup, Metlife,United Technologies and Wellpoint.9
What makes the Black-Scholes model the cornerstone of option valuation models is its ability to
figure a number of option prices in a short amount of time. However, one of the limitations of the
Black-Scholes model is that it values the option at only one point in timeat expiration.Therefore, this method is unable to calculate those options that have the option of early
exercise.10With that being said, the Black-Scholes model can be tweaked in an attempt tovalue American options. The Fischer Black Pseudo-American model can value these dividend-
paying options, but is less reliable for puts.11
Types of Volatility
As mentioned in the section above, the Black-Scholes model uses several parameters for
calculating the "fair" price of the option. Volatility (measured as a standard deviation) is the only
criterion that is not easily observed. Volatility represents the amount of uncertainty or risk that
lays in the underlying security.12It is one of the most common measures of risk. It measures
upside, as well as the downside, risk of investing in a particular financial instrument. Higher
volatility means that the price of the underlying security can rapidly change in either direction:
positive or negative.
Volatility can be used as a tool for trading. The greater the volatility, the more money making
opportunities exist. Short-term market players attempt to use volatile markets to make money, as
opposed to buying and holding strategies of the traditional" investors. Nowadays, volatility canbe traded directly, through derivative securities such as options and variance swaps.13
Four types of volatility exist: future, historical, implied, and seasonal.14The majority of traders
are interested in Future Volatility. Since no one can predict the future, most practitioners use
different methods for estimating volatility for the option pricing models by using Historical orImplied Volatility and adjusting for Seasonal Volatility for commodity trades.
Volatility can be estimated in several different ways. The most common measure of volatility is
the standard deviation of a return from the mean estimate. Using historical data, a trader can
calculate a Historical Volatility. Another method for assessing volatility incorporates various
option prices to estimate the stocks standard deviation and is calledImplied Volatility.
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The Black-Scholes model assumes constant volatility throughout the life of the option. While
using the model, an investor should use a forward-looking volatility estimate. Historical
Volatility is determined by looking back in time and using the historical standard deviation for
the future estimate. Implied Volatility is more reflective of the current market conditions and
shows an estimate of future underlying asset volatility that would produce the current marketvalue of the option(s). In the sections that follow, we will examine and compare various
volatilities of Caterpillar and John Deere stocks along with the S&P500 index to see how
different choices of volatility impact option pricing estimates.
Historical Volatility - HV
Historical Volatility (HV) uses the historical data to measure volatility of the financial
instrument. There are numerous ways to calculate Historical Volatility. The most common one,
and one of the more accurate ways is by computing the standard deviation of the logarithmicprice relatives.15
PR will stand for the price relatives (i.e. return for a period of time), then the formulas for the
mean and variance would be:
Mean
(4)
PR=1/Tt=1T(lnPRt)
where:
PR= mean price relativeT= number of annual time periods
PRt= price change in the underlying asset
Variance
(5)
VAR(PR)=1T1t=1T(lnPRtPR)2
where:
VAR(PR) = varianceT= number of annual time periods
PRt= price change in the underlying asset
PR= average price relative
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Standard deviation
(6)
(PR)=1T1t=1T(lnPRtPR)216
Historical Volatility is significantly impacted by two factors: the time period of observations and
the price relative intervals. Depending on the choice of these two parameters, the results can vary
widely. A practitioner has to take them into account as indicators of the range of possible
volatilities. Historical and Future Volatilities are sometimes referred to as Realized Volatility.17
Historical Volatility is evaluated and compared between Caterpillar and Deere. The goal is to
calculate the Historical Volatility for both companies and compare them quantitatively and
graphically. Data for each companies' stock prices was obtained from Google
Finance,http://finance.google.com/financefor the past 5-years from 12/6/02 to 11/30/07.Historical Volatility values are calculated using the equations shown above.
Two sets of volatility estimates were calculated for each of the companies. One estimate for the
entire 5-year span and one estimate for the most recent 1-year span:
Caterpillar Volatility Estimates:
12/6/02 to 11/30/07: (5-year span)
Standard Deviation: 3.79%
Annualized Standard Deviation (Volatility Estimate):27.34%
1/5/07 to 11/30/07: (1-year span)
Standard Deviation: 3.83%
Annualized Standard Deviation (Volatility Estimate): 27.60%
Deere Volatility Estimates:
12/6/02 to 11/30/07: (5-year span)
Standard Deviation: 3.80%
Annualized Standard Deviation (Volatility Estimate): 27.40%
1/5/07 to 11/30/07: (1-year span)
Standard Deviation: 4.37%
Annualized Standard Deviation (Volatility Estimate): 31.50%
http://finance.google.com/financehttp://finance.google.com/financehttp://finance.google.com/financehttp://finance.google.com/finance -
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Cannot fetch Flickr photo (id: 2091721660). The photo either does not exist, or is private Figure 2: Graph of Stock Comparisons 5-Years (Normalized)
Caterpillar and Deere
Cannot fetch Flickr photo (id: 2090969383). The photo either does not exist, or is private Figure 3: Graph of Stock Comparisons 1-Year (Normalized)
Caterpillar and Deere
Interpreting the Results:
By comparing the volatility estimations for Caterpillar, one can conclude that the volatility really
did not change between the 5-year span and the 1-year span. On the other hand, Deere's volatility
estimates were higher for the most recent 1-year span compared to the 5-year span. This means
that within the past 1-year, Deere's volatility increased.
By comparing the volatility estimates between Caterpillar and Deere's 5-year span, 27.34% and
27.40% respectively; one can conclude the volatility of both stocks remained fairly consistent.
This does not necessary mean that the stocks behaved similarly, it means that in either case, the
stock's probability of upward or downward movement are similar. On the other hand, by
comparing the 1-year span numbers: Caterpillar, 27.60% and Deere, 31.50%; one can conclude
that Deere had a higher probably of upward or downward movement in stock prices compared to
Caterpillar.
HV20 HV50 HV100 DATE CURIV Days/Percentile Close
STOCKHV2
0
HV5
0
HV10
0DATE
CURI
V
Days/Percentil
eClose
ONNOVEMBER 23, 2007
CAT 23 27 2707/11/2023
29.28 600/85%ile 68.63
DE 52 40 3607/12/2023
38.59 600/94%ile156.63
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S&P500(SPX)
24 24 1907/13/2023
22.86 600/97%ile144.07
ON
DECEMBER5, 2007
CAT 26 26 2607/11/2030
25.97 600/56%ile 71.88
DE 51 41 3707/11/2030
39.39 600/94%ile171.75
S&P500
(SPX) 26 20 21
07/11/203
0 22.70 600/96%ile
148.1
1
Table 1: Table of Volatilities as of November 23, 2007 and December 5, 200718
Where:
HV20: 20-day Historical Volatility - i.e. actual volatility
HV50: 50-day Historical Volatility
HV100: 100-day Historical Volatility
DATE: date of the last OPTION datacur_iv: the Implied Volatility of these options on DATE
Days: the number of days back for which Implied Volatility has been calculated
Percentile: measurement of the cur_iv, as compared to the past Days
Close: latest closing price of the underlying on November 23, 2007
Table 1 shows that Caterpillar's Implied Volatility is close to historical levels. A change from
29.28% to 25.97% in Implied Volatility values shows that it is not uncommon for CAT's IV to
change like this. Calculated by our team, Historical Volatility values are very close to Historical
Volatilities that are provided by Optionstrategist.com. The values of Historical Volatilitieschange depending on the time period, time interval, and exercise price of the options.
Since an observation has been made that Deere had a higher probability of movement compared
to Caterpillar in the past year, we should be able to make a similar observation on the price chart.
(See Figure 4)
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Two assumptions could be made to justify the higher volatility for Deere:
1. Overall, the slope of Deere's trend line is steeper, which yields a higher standarddeviation, therefore a higher volatility estimate.
2. Two significant spikes were seen in two relatively short time periods that are highlightedby the red circles below.
Cannot fetch Flickr photo (id: 2091120823). The photo either does not exist, or is private Figure 4: Graph of Stock Comparisons
1 year stock comparison of Caterpillar and Deere stock prices
An attempt to assess the accuracy of Historical Volatility: (Linear Regression)
Can one predict future stock prices by utilizing different regression techniques?
In order to assess the accuracy of Historical Volatility, a method has been developed to compare
the projection of Caterpillar and Deere's stocks via a regression line to the actual stock prices. A
total of two assessments will be conducted. 1.) Generating a linear regression line from year 1 toyear 4, and compare the year 5 projection to the actual year 5 prices shown in the red box (Figure
5) and 2.) Generating a linear regression line from year 3 to year 4, and compare the year 5
projection to the actual year 5 prices in the red box (Figure 6).
Cannot fetch Flickr photo (id: 2092009174). The photo either does not exist, or is private Figure 5: Graph of Caterpillar stock prices (5 years)
4-year linear regression model
Interpreting the above graph and regression line:
To further explain the above graph, you can see a regression line (blue) was calculated to reflectthe first 4 years of Caterpillar's stock prices (12/6/02 to last week of 06). The linear regression
line was then extended for another year for comparison of year 5 stock prices (shown in red box).
Cannot fetch Flickr photo (id: 2092009178). The photo either does not exist, or is private Figure 6: Graph of Caterpillar stock prices (2 years)
1-year linear regression model
Interpreting the above graph and regression line:
To further explain the above graph, you can see a regression line (blue) was calculated to reflect
the year 4 Caterpillar stock prices (2006). The linear regression line was then extended foranother year for comparison of year 5 stock prices (shown in red box).
Linear Regression Results
To start with, by looking at both linear regression charts above, you can clearly see the
regression chart with the smaller historical data did not predict the future stock prices as well as
the graph with the larger historical data regression line. In order to further analyze the accuracy
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of the regression estimates, the Mean Squared Error (MSE) was calculated for the projected
periods. To calculate the MSE, an investor first needs to take the square of the difference of the
projected price and the actual price over the sample period. Then, take the sum of all the squared
values and find the average.
(7)
MSE=t=1NEt2N
where:
N= the number of samples
Et= difference of the projected line and actual values
The MSE will ultimately indicate the accuracy of the regression model. A higher MSE means
that the prediction is poor, and a low MSE means that the prediction is more accurate.
The MSE for the 5-year sample is 51.86 and the MSE for the 2-year sample is 267.21. Since the
MSE for the 2-year sample is significantly higher than the 5-year sample, one can conclude that
the 5-year sample regression model is much more accurate compared to the 2-year model. This
result does not necessary mean the 5-year sample is truly accurate. It just means in this particular
case, the larger database yielded better regression predictions.
An attempt to assess accuracy of Historical Volatility: (Moving Average Method)
Linear regression models are not the only analysis tool to use in order to predict future stock
prices by using historical prices. Another tool is to use the moving average method. The moving
average method uses an average of recent prices to predict future prices. The below graph shows
how the moving average tool was used for Caterpillar for the last year of stock prices (2007).
Cannot fetch Flickr photo (id: 2092009182). The photo either does not exist, or is private Figure 7: Graph of Caterpillar stock prices (1 year)
1 and 2 month moving average models
To further clarify how the moving average is calculated for the chart above, the projection for the
first period is calculated by averaging the last 4 samples for the one month moving average
method and averaging the last 8 samples for the two month moving average method. Compared
to the linear regression models, you can clearly see the prediction with the moving average
method is much more accurate compared to the linear regression models. The comparison of
accuracy will be done once again by calculating the mean squared error.
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The MSE for the one-month moving average model is 10.92 while the two-month moving
average model is 17.36. Quantitatively, the accuracy for the moving average models are far more
accurate compared to the linear regression methods.
Historical Volatility charts provide useful information by giving a trader an idea about thehistorical levels of the volatility and risk in the underlying security and narrowing the range of
the parameter for the option pricing models. Reviewing the charts that are produced by various
tools online and by our simple model shows that Historical Volatility is not an exact number, but
rather a range that each investor should modify to their specific assumptions about the
fundamentals of the underlying asset and the market direction.
Charting the data is one of the best methods to quickly review this data. "Charts are not
mysterious, they are tools that allow investors to keep track of more opportunities & see quickly
when it's time to change strategies. They do not predict the future, but they are valuable in
determining the probably of success whether deciding to buy, to sell, or hold."19
When working with Historical Volatility measurements, the consideration should be made as to
using more data points vs. using more current data. The balance between the two is essential for
an accurate estimate of Historical Volatility.20
Also, Historical Volatility is used as an indicator of risk in the sense that stocks will require a
higher risk tolerance if the Historical Volatility is high.21In other words, stocks that have been
risky in the past have a greater chance of being risky in the future. Therefore, if an investor has
high risk tolerance, they would like to see a high level of volatility. An investor with a low risktolerance is liable to watch Historic Volatility closely in order to see which stocks have the
lowest rate.
As mentioned above, Historical Volatility works as a parameter in the Black-Scholes model
under the assumption that it is constant in time. However, in reality, volatility of the underlying
asset is constantly changing. Applying the value obtained from historical observations in the
model would not produce accurate results. Most practitioners are more concerned with
the estimatedorfuture volatility that would allow them to more accurately price the option
instead of having a range of Historical Volatilities. This is where Implied Volatility comes in andis used as additional information in the pricing model.
Implied Volatility - IV
According to Wikipedia, [I]mplied volatility of an option contract is the volatility implied bythe market price of the option based on an option pricing model. In other words, it is the
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volatility that, given a particular pricing model, yields a theoretical value for the option equal to
the current market price. The Implied Volatility is sometimes referred to as a measure of therelative value of the option.
It is the value of the volatility that makes the "fair" value of the option equal to its current price.Under the assumption that everybody in the market uses the same theoretical pricing model, such
as Black-Scholes, the discrepancy between the calculated value of the option and the market
value of the option is due to the difference in opinion about the inputs of the model .22
Using the previously discussed Black-Scholes model, the Implied Volatility can be determined
by knowing the following parameters for options with different strike price and maturity:
Current Stock Price Option Strike Price
Option Price Option Time to Maturity The Risk-Free Interest Rate
Once the parameters are substituted into the Black-Scholes model formula, the standard
deviation can be determined, which is the same as the Implied Volatility for an option. For the
majority of the stocks, several options with different expirations trade at the same time. Some
models take into account volatilities produced by each option to create weighted implied
standard deviations. Assumptions used in a model to average the volatilities impact the final
result. Most models give the highest weight to options that are the closest to "at-the-money"
positions. At-the-money options produce the least biased volatility estimates.
Numerous websites and programs provide Implied Volatility data to investors for a service fee or
for free. Optionstrategist.com is one of the sources that provide this information for free.
Looking again at thedata gathered from optionstrategist, the table shows that Implied Volatility
for both Caterpillar and Deere was different than any of the Historical Volatilities, but stayed
close to the range of historical values.
The phenomenon of the "Volatility Smile" is created and used to determine when the strike price
is in-the-money or out-of-the-money. The Volatility Smile occurs when a group of European
options with the same expiration date are graphed with Implied Volatility on the Y-Axis and
Strike Price on the X-Axis as shown below. It suggests that Black-Scholes under prices deep out-
of-the-money options as well as deep in-the-money options.
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Figure 8: The volatility smile
It would appear that the lowest possible Implied Volatility levels are options at-the-money. Inpractice, options are frequently sold by quoting the Implied Volatility versus the price.
Investopedia also suggests that, "A volatility smile is used by investors to price options in the
foreign currency market and the equity option market."23Theoretical values of the options
predicted with the Black-Scholes model are not 100% accurate because of the Volatility Smilephenomenon. An investor would use a "neutral, bull, or bear trading strategy" to make
money24taking into account an analysis of the fundamentals of the underlying asset.
Implied Volatility has been found to be an excessively unstable predictor of Realized Volatility.
For example, on the 3-month Eurodollar, Implied Volatility has been falling since 1985. At thesame time, interest rates and inflation have been declining while the credible relationship
between the Fed and foreign markets improve.25
Also, there is a direct correlation between significant events in economic history and Implied
Volatility. The largest changes in Implied Volatility occurred on the same days as the 1987
stock-market crash, the Persian Gulf War fears, and the debt crisis in Russia. The following table
shows the Top 20 changes in Implied Volatility and the economic news event that happened on
that day.
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Table 2: News Events that Coincide with Large Changes in Three-Month Eurodollar IV
26
This leads us to believe that those who work closely with Implied Volatility must be constantly
monitoring the global environment in order to stay ahead. If an investor can catch a big story
related to the economy, they may be able to react accordingly based on what will soon happen
with Implied Volatility.
As options become more and more popular in the market, understanding them has become even
more important. Historical Volatility has become a big focus in the understanding of options.
"The main manifestation of rising volatility is in increased prices for traded options. The prices
of call and put options traded on the London International Financial Futures Exchange (Liffe) are
derived from a complex formula, but one element is the Historical Volatility of the underlying
security."27
For those that often use derivatives in their work or for personal investments, it is essential to
understand Historical Volatility in order to make a profit. In contrast, Historical Volatility is
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often overlooked by long-term investors. They often see it as a nuisance and a waste of time
because it is not the true factor in price trends. However, Nick Louth of Financial Times feels
that long-term investors should not dismiss Historical Volatility. Improved understanding of the
types of volatility and methods of applying it to managing their portfolios should allow investor
make better decisions and improve profitability.
Volatility Indexes
This section looks at implied volatility through the volatility indexes of the S&P 500, NASDAQ,
and Dow Jones Industrial Average (DJIA). This gives us an opportunity to review how volatility
is measured in todays market and how it can impact a stock portfolio. We reviewed the threeindexes for the above markets: CBOE Volatility Index (VIX), the CBOE NASDAQ-100
Volatility Index (VXN), and CBOE DJIA Volatility Index (VXD). Next, we will investigate two
studies that discuss strategies on how to use volatility options to reduce a stocks portfolio riskwithout impacting the portfolios average return.
CBOEChicago Board Options Exchange
The Chicago Board Options Exchange (CBOEreferred to as the "See-bo") was founded in1973 and focuses on the option contracts for individual equities, indexes, and interest rates. It is
the worlds largest options market and is the market leader in developing new functionalproducts and technological innovations, specifically with electronic trading.28The VIX, VXN,
and VXD are products of the CBOE.
As defined, the CBOE "volatility indexes are key measures of market expectations of near-term
volatility conveyed by stock index option prices. The indexes measure the market's expectation
of 30-day volatility implicit in the prices of near-term index options. The indexes are quoted in
percentage points, just like the standard deviation of a rate of return. The indexes are leading
barometers of investor sentiment and market volatility relating to key stock indexes.29
The relationship between the volatility index and the stock index option price can be easily
observed through highly negative correlation, as can be seen in Figure 9. The correlation is the
measurement of the relationship between two variables. A coefficient of +1 means that thevariables move perfectly together. A coefficient of -1 means the variables move in the opposite
direction. The VIX is compared with the S&P 500 Index Options (SPX), the VXN is compared
with DJX - the options based on The Dow Jones Industrial Average (DJIA), and VXD is
compared with NASDAQ-100 Index Options (NDX). In the first half of 2007, the volatility
indexes all had negative correlations to the daily returns of the related stock indexes:
'VIXandSPX-0.86
http://www.cboe.com/micro/vix/introduction.aspxhttp://www.cboe.com/micro/vix/introduction.aspxhttp://www.cboe.com/micro/vix/introduction.aspxhttp://www.cboe.com/micro/spx/introduction.aspxhttp://www.cboe.com/micro/spx/introduction.aspxhttp://www.cboe.com/micro/spx/introduction.aspxhttp://www.cboe.com/micro/spx/introduction.aspxhttp://www.cboe.com/micro/vix/introduction.aspx -
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VXDandDJX-0.85
VXNandNDX-0.78'30
In addition, the correlations for years 2004 through 2006 have been graphed and show that the
correlation has been relatively consistent between the volatility index and stock index option:
Figure 9: Negative Correlations of the VIX and S&P 500 Indexes31
Figure 9 shows that when the stock index moves one way, the volatility index moves in the
opposite direction. This can be very helpful when attempting to reduce risk within a stockportfolio. This will be discussed in more detail later on.
VIX - CBOE Volatility Index
Figure 10: VIX Index32
The CBOE Volatility Index (VIX) shows the markets expectation of 30-day volatility. It isconstructed using the Implied Volatilities of the S&P 500 index options and is meant to be
forward looking and is calculated from both calls and puts. The VIX is a widely used measureof market risk and is often referred to as the "investor fear gauge" .33
http://www.cboe.com/micro/vxd/http://www.cboe.com/micro/vxd/http://www.cboe.com/micro/vxd/http://www.cboe.com/micro/djx/introduction.aspxhttp://www.cboe.com/micro/djx/introduction.aspxhttp://www.cboe.com/micro/djx/introduction.aspxhttp://www.cboe.com/micro/vxn/http://www.cboe.com/micro/vxn/http://www.cboe.com/micro/vxn/http://www.cboe.com/micro/ndx/introduction.aspxhttp://www.cboe.com/micro/ndx/introduction.aspxhttp://www.cboe.com/micro/ndx/introduction.aspxhttp://www.cboe.com/micro/ndx/introduction.aspxhttp://www.cboe.com/micro/vxn/http://www.cboe.com/micro/djx/introduction.aspxhttp://www.cboe.com/micro/vxd/ -
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The VIX was introduced in 1993 by the CBOE and was a weighted measure of the ImpliedVolatility of eight S&P 100 at-the-money put and call options.34The calculation for the VIXwas based on the Black-Scholes pricing model and, from the beginning, was considered to be
'the world's premier barometer of investor sentiment and market volatility'. It is widely followed
and has been cited in hundreds of news articles in the Wall Street Journal, Barron's, and otherleading financial publications.35
In 2003, a more robust methodology for the VIX was introduced. The fundamental features of
the VIX remain the same and it continues to provide a minute-by-minute snapshot of expectedstock market volatility over the next 30 calendar days.36In the opinion of many, this allows fora more accurate view of investors expectations on future market volatility. VIX values greaterthan 30 are generally associated with a large amount of volatility as a result of investor fear or
uncertainty, while values below 20 generally correspond to less stressful, even complacent, times
in the markets.37
According the CBOE, the most significant change is a new method of calculation. The newVIX estimates expected volatility from the prices of stock index options in a wide range of strike
prices, not just at-the-money strikes as in the original VIX. Also, the new VIX is not calculated
from the Black-Scholes option pricing model; the calculation is independent of any model. The
current VIX uses a newly developed formula to derive expected volatility by averaging the
weighted prices of out-of-the-money puts and calls. This simple and powerful derivation is based
on theoretical results that have spurred the growth of a new market where risk managers and
hedge funds can trade volatility, and market makers can hedge volatility trades with listed
options."38
The formula used to determine the current VIX calculation is:
Where
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39
The following link provides a detailed explanation on how the VIX is calculated:VIX
Calculation
The second significant change in the current calculation of VIX is that it now uses the options of
the S&P 500 instead of just the S&P 100. According to the CBOE, The S&P 500 is the primaryU.S. stock market benchmark and the reference point for the performance of many stock funds.
In addition, the S&P 500 underlies the most active stock index derivatives, and it is the domestic
index tracked by volatility and variance swaps.40
The updated VIX is considered to be more accurate than before and is now measured in the same
way that financial theorists, risk managers, and volatility traders have come to measure it. In
addition, the current VIX is considered more robust because it pools the information fromoption prices over the whole volatility skew, not just from at-the-money options."41
As discussed earlier, the relationship between the CBOE Indexes and the related stock indexes
are negatively correlated. This holds true for the VIX and S&P 500 Index as well.
http://cfe.cboe.com/education/vixprimer/About.aspxhttp://cfe.cboe.com/education/vixprimer/About.aspxhttp://cfe.cboe.com/education/vixprimer/About.aspxhttp://cfe.cboe.com/education/vixprimer/About.aspxhttp://cfe.cboe.com/education/vixprimer/About.aspxhttp://cfe.cboe.com/education/vixprimer/About.aspx -
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Figure 11: Negative Correlation of the Original VIX, the New VIX and the S&P 500
Indexes-4 Months42
The VIX is called the 'investor fear gauge' for a reason. Given the strong negative correlation
with the SPX, there are many points in history where drastic increases with one, correspond with
a significant decline in the other and vice versa. Specifically, this can be seen during both Gulf
Wars. These were times of extremely high volatility in the market where the VIX increased
drastically as the SPX declined just as drastically.
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43Figure 12: Negative Correlation of the Original VIX, the New VIX and the S&P 500
Indexes13-Year Span with Significant Events
VXN - CBOE NASDAQ-100 Volatility Index
Figure 14: VXN Index44
The CBOE NASDAQ-100 Volatility Index (VXN) is "the measure of the implied volatility for
the NASDAQ 100".45The VXN is calculated using the same formula and methodology that is
used to calculate the current VIX. As defined by the CBOE, "It measures the marketsexpectation of 30-day volatility implicit in the prices of near-term NASDAQ-100 options. VXN
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is quoted in percentage points, just like the standard deviation of a rate of return."46Similar to the
VIX, the VXN is the investor fear gauge for the NASDAQ-100 Index (NDX). It gauges the
investor sentiment and market volatility related to the NDX.47
Figure 15: NDX and VXN Indexes48
VXD - CBOE DJIA Volatility Index
Figure 16: VXD Index49
The CBOE DJIA Volatility Index (VXD) was introduced in 2005 and "tracks the volatility of the
Dow Jones Industrial Average (DIJA) by measuring Implied Volatility of the near-term DJX
options". It is designed to reflect investors' consensus view of expected volatility over the next
30 days in the DJIA, so like the other CBOE Indexes, it can be used as "a benchmark of investor
sentiment".50
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Figure 17: CBOE DJIA Volatility Index (VXD)51
Figure 18: DJX and VXD Indexes52
VIX, VXN , VXD Option Strategies
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Options for the VIX, VXN, and VXD are types of non-equity options that use the same specific
CBOE Volatility Index as the underlying asset. This gives individual investors the ability to trade
market volatility in order to try to reduce the risk of their overall portfolio. Trading CBOE
options can be a useful tool for investors wanting to hedge their portfolios against sudden market
declines, as well as to speculate on future moves in volatility.53
In 2007, the Fund Evaluation Group (FEG) completed a new study entitled "Evaluation of
BuyWrite and Volatility Indexes - Using the CBOE DJIA BuyWrite Index (BXD) and the CBOE
DJIA Volatility Index (VXD) for Asset Allocation and Diversification Purposes." The paper
covers a 9-year period from October 1997 to November 2006 and presents one strategy for
reducing risk in a portfolio using VXD options. The study concluded that "a small (10%)
allocation to the CBOE DJIA Volatility Index (VXD) could have reduced the volatility of an all-
stock portfolio by about 26%, without materially affecting returns". Likewise, as the graph below
illustrates, an almost optimal risk-return investment strategy would be an allocation of
approximately 20% to the VXD. This strategy would optimize the overall portfolio return, but at
the lowest possible risk. Another advantage found was how the VXD option reacted to
increasing and declining markets. "This showed that VXD increased more during market
declines (VXD reacted more to stock market declines than to stock market advances), indicating
that VXD has potential as a diversification tool." Likewise, "the inclusion of a small (5%)
allocation to the VXD Index Option boosted risk-adjusted returns for a stock-oriented portfolio,
and lowered the risk-adjusted returns for a fixed-income-oriented portfolio."54
http://www.cboe.com/micro/vxd/FEG_Paper_Jan23_Final_BXD_VXD.pdfhttp://www.cboe.com/micro/vxd/FEG_Paper_Jan23_Final_BXD_VXD.pdfhttp://www.cboe.com/micro/vxd/FEG_Paper_Jan23_Final_BXD_VXD.pdfhttp://www.cboe.com/micro/vxd/FEG_Paper_Jan23_Final_BXD_VXD.pdfhttp://www.cboe.com/micro/vxd/FEG_Paper_Jan23_Final_BXD_VXD.pdfhttp://www.cboe.com/micro/vxd/FEG_Paper_Jan23_Final_BXD_VXD.pdfhttp://www.cboe.com/micro/vxd/FEG_Paper_Jan23_Final_BXD_VXD.pdfhttp://www.cboe.com/micro/vxd/FEG_Paper_Jan23_Final_BXD_VXD.pdf -
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Figure 19: Efficient Frontier
55
A recent article from Barron's, Modern Portfolio Protection by Lawrence G. McMillan, discusses
a similar strategy regarding the use of the VIX option. McMillan discusses using VIX options as
an insurance policy. He explains, "The most popular form of "macro" protection has been buyingout-of-the-money SPX puts, but a new class of derivatives, based on the CBOE Volatility Index,
or VIX, has grown in popularity." "Buying volatility futures, or call options on volatility,protects against sharp increases in volatility, which typically occur when the stock market drops.
VIX calls are a better hedge for a broad-based equity portfolio than SPX puts, because they
provide dynamic protection." Also, according to McMillan, the cost of using a VIX option is
lower than that of a SPX put. He explains, "Also, owing to the extreme volatility of the VIX, you
need only protect about 10% of the value of a stock portfolio, thereby keeping the overall cost of
this insurance lower than that of protection using SPX puts."56
As we have seen from both studies, the market volatility indexes can be used to not only measure
risk, but also to reduce the overall risk of a stock portfolio. A relatively small percentage of ones'
overall stock portfolio can be invested in options in order to help reduce the overall risk of the
portfolio.
Conclusion
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Volatility represents the amount of uncertainty or risk that lays in the underlying security. It is
one of the most common measures of risk. This report looked at the practice of using Historical
Volatility data and Implied Volatility data to measure volatility in order to predict performance.
By analyzing the different methodologies of measuring volatility, we set out to determine which
method more accurately forecasts the actual volatility and in which conditions it works best.
The Black-Scholes Option Pricing Model is an easy way to value options, and is widely used and
accepted by large companies and major investors. However, it has the limitation of only
providing a value for the option at the expiration of the option (European Style). For an
American option, the Fischer Black Pseudo-American model can be used, but it is better applied
to calls as it is less reliable for puts.
Historical Volatility looks back in time and uses the historical standard deviation for future
estimates. It is significantly impacted by the time period of the observations as well as the price
relative intervals. So, the results can vary widely depending upon the choice of these two
parameters and ultimately upon the knowledge and skill of the investor making these choices.
Implied Volatility is more reflective of the current market conditions than Historical Volatility,
and provides an estimate of the future underlying asset volatility that would produce the current
market value of the option. Again, the knowledge and skill of the investor plays into the
determination of Implied Volatility because discrepancies between the calculated value of the
option and the actual market value of the option are due to the difference in opinions about the
inputs of the model.
Our analysis of the volatility of Caterpillar and Deere showed that Implied Volatility was
different than any of the Historical Volatility values, but did stay close to the range of historical
values. Therefore, for the time period of our study, we can say that the Historical and Implied
Volatilities were about the same.
Other works have found that for the three-month Eurodollar, Implied Volatility has been an
excessively unstable predictor of Realized Volatility. However, those that realize the relationship
between current events and Implied Volatility can arbitrage if they can be the first to learn of
major current events and react accordingly with respect to the Implied Volatility calculatedbefore and after breaking news.
Pricing options in the foreign currency market and the equity option market can be done with the
use of the Volatility Smile. The Volatility Smile is used to determining when the strike price is
in-the-money or out-of-the-money. However, it can under price deep out-of-the-money options
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as well as deep in-the-money options. Therefore, investors should use the appropriate neutral,
bull, or bear trading strategy and analyze the fundamentals of the underlying asset.
The previous strategies are applied generally for individual stocks, bonds, and portfolios.
However, the volatility options can be used to reduce a stock portfolios risk without loweringthe portfolios average return. Since volatility indices are key measures of market expectations ofnear-term volatility conveyed by stock index option prices, the VIX, VXN, and VXD are leading
barometers of investor sentiment and market volatility relating to key stock indexes. The
correlation between the volatility index and the stock index is highly negative. Therefore, when a
stock index moves one way, the volatility index moves in the opposite direction. This gives
investors the ability to trade market volatility in order to try to reduce the risk of their overall
portfolio. Also, VIX, VXN, and VXD options can be useful tools for investors wanting to hedge
their portfolios against sudden market declines, as well as to speculate on future moves in
volatility.
Ultimately, the knowledge, skill, and risk aversion of the investor will factor into the decisions
and trading style that will be undertaken by the investor. While Historical Volatility has been
shown to work well in the long run, Implied Volatility can be utilized better in the short term,
especially when an investor has the latest news and current events to consider. The best approach
seems to be a blend of both Historical and Implied Volatility with a good understanding of the
underlying assets.
Bibliography
Footnotes
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host-live3. Unknown Author. "Stock Price Dynamics." OS Financial Trading System. 1999. 12 Dec. 2007.4. Unknown Author. "Stock Price Dynamics." OS Financial Trading System. 1999. 12 Dec. 2007.5. Risk Glossary, Encyclopedia & Resource Locator. "Black-Scholes (1973) Option PricingFormula". Riskglossary.com. 1996. 12 Dec.
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2007http://www.riskglossary.com/link/black_scholes_1973.htmNov 23, 20076. Mark Grinblatt and Sheridan Tittman: Markets and Corporate Strategy2nd. McGraw-Hill,2004. 284-2857. Mark Grinblatt and Sheridan Tittman: Markets and Corporate Strategy2nd. McGraw-Hill,
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treat it as a nuisance; it can bring in a profit if used the right way" Jan 11, 2003, pg 7, Dec5, 2007.28. Investopedia.http://www.investopedia.com/terms/c/cboe.asp; Dec 8, 2007
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